In mathematics, **sexy primes** are prime numbers that differ from each other by six. For example, the numbers 5 and 11 are both sexy primes, because 11 - 5 = 6.

The term "sexy prime" is a pun stemming from the Latin word for six: *sex*.

If *p* + 2 or *p* + 4 (where *p* is the lower prime) is also prime, then the sexy prime is part of a prime triplet.

## Contents

## n# notation

As used in this article, *n*# stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ *n*.

## Types of groupings

### Sexy prime pairs

The sexy primes (sequences OEIS: A023201 and OEIS: A046117 in OEIS) below 500 are:

- (5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107), (103,109), (107,113), (131,137), (151,157), (157,163), (167,173), (173,179), (191,197), (193,199), (223,229), (227,233), (233,239), (251,257), (257,263), (263,269), (271,277), (277,283), (307,313), (311,317), (331,337), (347,353), (353,359), (367,373), (373,379), (383,389), (433,439), (443,449), (457,463), (461,467).

As of May 2009^{[update]} the largest known sexy prime was found by Ken Davis and has 11,593 digits. The primes are (*p*, *p*+6) for

*p*= (117924851 × 587502 × 9001# × (587502 × 9001# + 1) + 210) × (587502 × 9001# − 1)/35 + 5.^{[1]}

9001# = 2×3×5×...×9001 is a primorial, i.e., the product of primes ≤ 9001.

### Sexy prime triplets

Sexy primes can be extended to larger constellations. Triplets of primes (*p*, *p* + 6, *p* + 12) such that *p* + 18 is composite are called **sexy prime triplets**. Those below 1000 are (OEIS: A046118, OEIS: A046119, OEIS: A046120):

- (7,13,19), (17,23,29), (31,37,43), (47,53,59), (67,73,79), (97,103,109), (101,107,113), (151,157,163), (167,173,179), (227,233,239), (257,263,269), (271,277,283), (347,353,359), (367,373,379), (557,563,569), (587,593,599), (607,613,619), (647,653,659), (727,733,739), (941,947,953), (971,977,983).

As of 2013^{[update]} the largest known sexy prime triplet, found by Ken Davis had 5132 digits:

*p*= (84055657369 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1) / 35 + 1.^{[2]}

### Sexy prime quadruplets

Sexy prime quadruplets (*p*, *p* + 6, *p* + 12, *p* + 18) can only begin with primes ending in a 1 in their decimal representation (except for the quadruplet with *p* = 5). The sexy prime quadruplets below 1000 are (OEIS: A023271, OEIS: A046122, OEIS: A046123, OEIS: A046124):

- (5,11,17,23), (11,17,23,29), (41,47,53,59), (61,67,73,79), (251,257,263,269), (601,607,613,619), (641,647,653,659).

In November 2005 the largest known sexy prime quadruplet, found by Jens Kruse Andersen had 1002 digits:

*p*= 411784973 · 2347# + 3301.^{[3]}

In September 2010 Ken Davis announced a 1004-digit quadruplet with *p* = 2^{3333} + 1582534968299.^{[4]}

### Sexy prime quintuplets

In an arithmetic progression of five terms with common difference 6, one of the terms must be divisible by 5, because 5 and 6 are relatively prime. Thus, the only sexy prime quintuplet is (5,11,17,23,29); no longer sequence of sexy primes is possible.

## See also

- Cousin prime (two primes that differ by 4)
- Prime k-tuple
- Twin prime (two primes that differ by 2)

## References

**^**Ken Davis, "11,593 digit sexy prime pair". Retrieved 2009-05-06.**^**Jens K. Andersen, "The largest known CPAP-3". Retrieved 2014-06-13.**^**Jens K. Andersen, "Gigantic sexy and cousin primes". Retrieved 2009-01-27.**^**Ken Davis, "1004 sexy prime quadruplet". Retrieved 2010-09-02.

- Weisstein, Eric W. "Sexy Primes".
*MathWorld*. Retrieved on 2007-02-28 (requires composite*p*+18 in a sexy prime triplet, but no other similar restrictions)

## External links

- Grime, James. "Sexy Primes (and the only sexy prime quintuplet)".
*Numberphile*. Brady Haran.